Simulation and Results for µ controller

The robust computation for a stabilising controller shows that µ-synthesis, but not H∞ control, can guarantee the stability and performance robustness; therefore,

µ-synthesis is now further tested in a nonlinear environment. In this study, a steady

atmosphere is assumed, and the effect of wind is neglected. The strength of the controller is tested against a wider region around the trimming equilibrium point in order to cover a major part of the flight envelope.

Cruising Case

Because it is the largest segment of flight, the cruise case is analysed first. To test the performance of the controller in the time domain, the bank angle signal reference

input was assumed to be: ϕ(t) =                    7.5 deg/s 2<t<4 15 deg 4<t<34 −7.5 deg/s 34<t<36 0 otherwise

Table 5.6 lists the testing conditions around the trimming point. The results of Table 5.6 Trim and testing operating conditions.

Condition Altitude, m Mach Speed, km/h Trimming point 4000 0.3 367 Test condition 1 3475 0.26 318 Test condition 2 3750 0.28 342 Test condition 3 4266 0.32 392

nonlinear testing are shown in Figure 5.18. With a 30% faulty aileron, the nonlinear

Fig. 5.18 Nonlinear simulation of bank (rolling) angle ϕ at cruise.

test of the entire system showed good consistency between the linear and nonlinear controlled closed-loop models. Moreover, from a control design viewpoint, the robust

µ-synthesis controller led to a good tracking response of the bank (rolling) angle ϕ, where the steady-state error was very small. The aircraft response was slightly

sluggish (rise time, 11 s) owing to the effects of the fault. However, the robust controller maintained good performance under different operating conditions.

5.4 Non-linear Testing and Validation 101

Fig. 5.19 Aileron (ξ) and rudder (ζ) deflections to generate the required response.

For control surfaces deflections shown in Figure 5.19, the aileron actuates smoothly at a maximum rate of 5 deg/s and a deflection of 32 deg. However, the deflection was high when testing at low speed (0.26 Mach).

The phase and gain margins indicate that there is still a safety margin before instability. The feedback closed loop system between ϕref and ϕactual is expressed by:

ϕref

ϕactual

= L = GK 1 + GK

where G is the aircraft lateral dynamical model and K is the µ-controller. Results as shown in Figure 5.20 are: the gain margin GM= 17.7 dB @ 0.582 rad/s and the phase margin PM= 117 deg. @ 0.129 rad/s. This is also confirmed by the Nyquist

plot as in Figure 5.21, where the gain margin is determined by the expressions below. From the Nyquist plot, L(jωp) is found as shown in Figure 5.22. Then the gain

Fig. 5.21 Nyquist plot for stability.

Fig. 5.22 Computing GM using Nyquist plot.

margin is:

GM = 20log₁₀ 1

|L(jωp)|

L(jωp)= 0.13, thus, GM=17.7 dB. Phase margin is angle between 0 and where

the plot crosses the unity circle in the counter clockwise direction, which is equal to PM= 117 deg.

5.4 Non-linear Testing and Validation 103

Ascending Case

Fig. 5.23 Nonlinear simulation of bank angle ϕ under ascent operating conditions.

For the ascending scenario, the bank angle reference signal was requested, and the aircraft response was analysed accordingly. Figure 5.23 shows the testing results for

Fig. 5.24 Nonlinear simulation of aileron deflection angle ξ under ascent operating conditions.

the bank angle response at different operating points around the trimming condition. First, the slow response and overshoot were due to the 30% fault in the aileron, which limited the control power. However, the response was stable and converging, and the controller maintained stability under different ascending trimming conditions. In

Figure 5.24, the aileron deflection is analysed. At very low speeds (0.16 and 0.18 Mach), the deflection was high. This was because the moments generated by the actuators depend on the aircraft’s speed squared; so at low speed, high deflection of the actuators was required. In addition, the test showed that the linear model has no actuator dynamics, resulting in a non-oscillatory deflection pattern compared to the nonlinear response.

Descending Case

Finally, the last testing case is the aircraft descent. The response of the bank angle

ϕ (Figure 5.25) shows an acceptable response for the linear and nonlinear cases

in terms of the overshoot, stability, and settling error. For the aileron deflections

Fig. 5.25 Nonlinear simulation of bank angle ϕ under descent operating conditions.

(Figure 5.26), the maximum limit and rate of change were acceptable. Moreover, the controller asymptotically stabilised the system under different operating conditions even with a 30% actuator loss-of-efficiency fault.

5.5 Conclusion 105

Fig. 5.26 Nonlinear simulation of aileron deflection angle ξ under descent operating conditions.

5.5

Conclusion

In this chapter, an optimal fault tolerant controller was developed for a flying vehicle where the robust features of the H∞ and µ-synthesis techniques were utilised. The

work contained modelling of actuator loss of efficiency faults as an uncertainty in the system, which can be handled inherently with these control techniques. Thus, this type of method does not rely on any fault detection schemes. Robust computation involved the minimisation of a cost function that takes into account multiple design objectives. These objectives are the tracking error minimisation, control energy minimisation, and uncertainties. The modelling of faults as an uncertainty in the system enabled the author to exploit the robustness of the controllers and test the control law for aircraft subjected to actuator faults. The weighting functions for optimisation objectives are selected and designed to shape the corresponding relevant requirements. When the linear-based optimised solutions were found, the system was tested in the nonlinear environment for more practical validation. Three design points were chosen within the flight envelope, which are the ascending, cruising, and descending operating cases. At these selected flying scenarios, the robust optimal fault-tolerant controllers were tested.

Results showed that the H∞ baseline controller managed to give good tracking

and control input performance for the multi-variable system. However, it could not withstand against actuator faults. The system’s deflection performance was deteriorated by very high-frequency, rapid fluctuations, and sharp deflections as well. On the other hand, in all flying cases, when testing the µ-synthesis-based controller, the system maintained great tracking performance and smooth rate of change control deflections with 30% fault magnitudes.

Moreover, the system was tested at operating points near to the trimming points to test the region around them, which can be covered by one controller. Results showed that when the testing point is at lower speed, the robust µ-synthesis controller maintained the tracking performance, but led to higher aileron deflections. The gain and phase margins for the system were computed in the cruising case and results showed that the system has acceptable margins with the robust controller before it becomes unstable.

Further work may include further optimisation of the µ-synthesis-based controller that may lead to the increase in the fault-tolerance degree. This could be by the tuning of control or tracking weighting functions or the way the optimisation problem is solved. Another extension is the lock in place fault accommodation problem, which may need a considerably reconfigured control law.

Chapter 6

Reconfigurable Dynamic Control

Allocation against Lock in Place

Faults

6.1

Introduction

The primary means of control surfaces used to control the rolling, pitching, and yawing motion of the aircraft are the ailerons, elevators, and rudder, respectively. When one of the actuators incurs a loss of effectiveness fault, the adaptation or robustness of the controller could enhance the controllability and handle the situation by maintaining minimum stability requirements, as discussed in the previous chapters. Now, a more sophisticated problem, when one actuator experiences a lock in place fault, is investigated. This type of fault usually occurs due to, for instance, a structural jam that prevents the actuator from moving, or hydraulic oil leakage. etc. [110]. The lock in place faults will certainly introduce undesirable adverse moments that divert the aircraft out of its trajectory. In the passive fault-tolerant control techniques, discussed in Chapter 5, deviations of the plant parameters or the actuators from their expected position were accommodated by a fixed robust feedback controller that accounts for uncertainties up to a certain limit. However, if these deviations became excessively large and exceed the robustness properties, other actions need to be taken. Therefore, an active fault-tolerant control approach is proposed here. The problem includes the exploitation of the other available control effectors, which needs a thorough investigation to resolve this critical problem automatically. For these jam actuator effects, functional redundancy is considered. Typically, it will be assumed that the redundant hardware that powers the faulty actuator are unusable, thus, it is the role of the other healthy actuators to cope with this situation. In

flight control real applications, the redundancy is extended to cover the aerodynamic control effectors. For example, the Boeing 747 has a three-segment rudder, where the redundant control system’s channels independently power each segment [11]. In this work, the utilisation of healthy effectors to handle lock in place faults is considered for automatic closed loop operation. An advanced optimal multivariable approach is exploited for this fault tolerant control problem, which is the control allocation. In recent years, flight control design techniques that compute the moments to be produced in pitch, roll, and yaw orientations, rather than the control surface deflections, have gained increased attention. These methods generate the virtual, or generalized, control commands v(t), as shown in Fig. 6.1, that are transferred and allocated to the aircraft’s control surfaces. The problem of transforming these virtual control commands is commonly known as the control allocation problem [111].

Fig. 6.1 Control allocation generic scheme

Thus, the main objective of control allocation is to compute a control input u(t) that ensures that the commanded virtual control v(t) is cooperatively produced by all effectors [112]. The control allocation (CA) approach has the following features [113], [110]:

1. It handles the redundancy of the control surfaces for certain optimized objec- tives, such as trajectory following and surface deflection prioritization, etc. 2. Actuator constraints, such as rate and position limits, are taken into account

in the optimisation solution. When one actuator is saturated, the other control surfaces can produce the required control efforts.

3. The allocation problems, which are based on optimization techniques, are emphasized since their computational requirements are within the capabilities of current embedded computer technology.

The principles of control allocation, in general, are not restricted to motion control systems. Therefore, virtual control command v(t) is not limited to representing the generalised forces or moments, but could also represent quantities such as mass and